Model of phase fluctuations in a latticed-wave superconductor: Application to the Cooper-pair charge-density wave in underdoped cuprates

Abstract

We introduce and study an $XY$-type model of thermal and quantum phase fluctuations in a two-dimensional correlated lattice $d$-wave superconductor based on the ${\mathrm{QED}}_{3}$ effective theory of high-temperature superconductors. General features of and selected results obtained within this model were reported earlier in an abbreviated format (Z. Te\ifmmode \check{s}\else \v{s}\fi{}anovi\ifmmode \acute{c}\else \'{c}\fi{}, e-print cond-mat/0405235). The model is geared toward describing not only the long distance but also the intermediate length-scale physics of underdoped cuprates. In particular, we elucidate the dynamical origin and investigate specific features of the charge-density wave of Cooper pairs, which we argue is the state behind the periodic charge-density modulation discovered in recent scanning-tunneling-microscopy experiments. We illustrate how Mott-Hubbard correlations near half-filling suppress superfluid density and favor an incompressible state which breaks translational symmetry of the underlying atomic lattice. We show how the formation of the Cooper pair charge-density wave in such a strongly quantum fluctuating superconductor can naturally be understood as an Abrikosov-Hofstadter problem in a type-II dual superconductor, with the role of the dual magnetic field played by the electron density. The resulting Abrikosov lattice of dual vortices translates into a periodic modulation of the Bogoliubov--de Gennes (BdG) gap function and the electronic density. We numerically study the energetics of various Abrikosov-Hofstadter dual vortex arrays and compute their detailed signatures in the single-particle local tunneling density of states. A $4\ifmmode\times\else\texttimes\fi{}4$ checkerboard-type modulation pattern naturally arises as an energetically favored ground state at and near the $x=1∕8$ doping and produces the local density of states in good agreement with experimental observations. The leading-order behavior of nodal BdG fermions remains unaffected.